Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig relations

Abstract We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey Kramers-Kronig relations. The apparent contradiction with the principle of causality is also clarified. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive Kramers-Kronig analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical Hamiltonian systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. In order to gain a more complete picture, connecting the response theory for equilibrium and non equilibrium systems, we show how to rewrite the classical response theory by Kubo for systems close to equilibrium so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. Finally, we briefly discuss how the presented results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might have relevant implications for climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, Kramers-Kronig relations might be more robust tools for the definition of a self-consistent theory of climate change.